Topic 4

Question 1

What are real-world examples of Quantum Wells which are finite?

Answer 1

A sandwich of different sorts of semiconductor material.

These material which are very wide in y and z, along x it is narrow.

We can model this with a finite potential.

In one layer we have a certain potential. In the layers on either side of
we have a different much larger potential.

Electrons in the well can quantum tunnel.

Question 2

Consider a particle in a finite potential well.

Answer 2

A particle of energy less than the height of the well (E

Question 3

The particle is in a symmetrical well

Answer 3

There will be even parity and odd parity.

Consider the BC of each wall [the wave function must be a continuous function of x and the slope of the function in x must be continuous.] [One value at the walls]

Even parity solutions are only Bcoskx

The condition of continuity at the boundaries occurs when the general solution is equated to each other

The slope of the wave function divided by the wave function gives condition for a smooth join at x which can only be satisfied
when the parameters k_0 and α take on special values[NOT INDEPDENTENT FROM EACH OTHER].
tan(kL/2)=α/k
Split into two equations and introducing a new constant k_0 and θ_0. Obtaining two simultaneous equations
α^2+k^2=k_0^2 andα^2+θ^2=θ_0^2

Then solve this geometrically by seeing where y=tanθ intersects with
y=α/k [same for odd parity but odd parity solutions are only Bsinkx and cot is used and -α/k]

Question 4

What are the general properties of solutions to the FPW

Answer 4

• larger V0 ⇒ more bound states; smaller V0 ⇒ less bound states
• there is always at least one (symmetric) bound state, even in a very shallow well (V0 & 0)
• the wavenumber and energy of the nth state is less than in the IPW for which the wavenumber [intersection with alpha/k line is always a bit less] [Wavefunction spreads out by entering the classically forbidden region so KE reduces]